Filtrations on the Chow groups and transcendence degree

Details Filtrations on the Chow groups and transcendence degree Filtrations on the Chow groups and transcendence degree

2002-12-04

arxiv.org/abs/math/0212057v2

Mathematics

Algebraic Geometry

12 pages, case of higher graded pieces is treated

Scientific paper

For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending of the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of transcendence degree one. This may be viewed as a refinement of results of Nori, Schoen, and Green-Griffiths-Paranjape. For higher graded pieces we have a similar assertion assuming a conjecture of Beilinson and Grothendieck's generalized Hodge conjecture.

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